Computational Design of BCH-Codes and Their Applications in the Data Security

Abstract

Due to innovations in communication technologies, digital medium is extensively used across the World. Large amount of information in digital form is stockpiled in digital libraries. The error free transpOl1ation of digital data through untrustworthy channels is a great challenge. Accordingly, in information theory, telecommunication, computer science and algebraic coding theory, an error correction code or error correcting code is used for controlling errors in data over unreliable or noisy communication chalmels. Because of error correcting codes, the communication is made over the shol1 and long distances without any obstacle. Thus, it made possible the gigabit data transmission over the wireless communication mediums. Indeed, it is the fundamental part of the modern communication systems and essentially utilized in hardware level implementations of intelligent and smart machines like telecom equipment, highly sensitive video cameras, optical devices, and scanners. Development of data transferring codes were started with the first article [3 J] of Claude Shannon in 1948. He explained that, every communication channel has some capacity. If the rate of data transmission is smaller than capacity, then design of communication system for the channel is possible with the help of data transmission codes. This system has least probability of output errors, but Shannon did not give the method for the construction of such type of codes. In 1950, for this purpose Hamming [14] and Golay [9] introduced cyclic block codes known as binary hamming and Golay codes respectively. These classes of codes have the capability to detect up to two errors and correct one error. Furthermore, these codes have fascinating features and can be easily encoded and decoded but are not suitable for multiple errors. In 1953, Muller [18] introduced a mUltiple error correcting codes technique and Reed [26] developed decoding technique of such type of codes. Yet, Shannon's hypothesis remained ulU'esolved. Cyclic codes are one of the dynamic class of error correcting codes. In 1957, Prange [33] initiated an idea of cyclic codes in two symbols. In addition, Prange [24] used the coset equivalence for decoding the group codes in 1959. After that, a big development in the theOlY of cyclic codes was made to correct burst along with random errors initiated by various researchers. The cyclic codes were initially developed over binary field ::l2 and into its Galois field extension GF(Z7n). Though, it was further extended over the prime field ::lp and into its Galois field extension GF(p7n). The remarkable development in coding theory began when Hocquenghem [10], Bose and Chaudhuri [3] explained the large class of codes which correct multiple errors known as BCH codes in 1960. They explained the BCH codes over Galois field. These codes are generalization ofbinalY Hamming codes. The advantage ofBCH code is that, Fundamentally the BCH codes are utilized for only data transmission, but not for data security. In this study we have given the idea that, BCH codes can be used for data security. Accordingly, by BCH codes over Galois field and Galois ring a couple of techniques are devised to modifY AES algorithm. Accordingly, this modified AES algorithm tested on text and image data, the results assured the appropriate level of security. This thesis consists of seven chapters. In Chapter one, some important notions of algebraic structures and error correcting codes are explained which are necessary for understanding further chapters. In Chapter two, initially we have given details on obtaining the maximal cyclic subgroup of group of units of a Galois ring through computational method. Afterword the new computational encoding scheme of BCH code over Galois ring is introduced. This novel computational approach of encoding of BCH codes provides generator polynomial for any length n corresponding to each designed distance d. Furthermore, the encoding ofBCH codes over Galois field has also been explained with the help of reduction map. Another outcome of this study is that one can find the dimensions of primitive BCH codes for any length and designed distance. In Chapter three, using C# computer language a computational decoding scheme for BCH codes over Galois ring has been designed by which Barlekamp Massey decoding algorithm of BCH codes over Galois field is employed to correct the errors. Indeed, this modified Barlekamp Massey decoding algorithm is designed for large length BCH codes over Galois field. The special feature of this study is the syndrome calculation with computational approach. Thus, decoding of BCH and RS codes over Galois ring by using modified Barlekamp Massey algorithm has been ensured. In Chapter four, BCH codes have been utilized to improve the AES algorithm. BCH codes have been utilized as a secret key in round key addition step of AES algorithm. ill addition, using BCH codes, the maximum distance separable matrix has been constructed and applied in mixed column matrix step in modified AES algorithm. Thus, this modified AES algorithm has been applied in image encryption and different analyses on encrypted image have been performed. The comparison of results of encrypted image by using original and modified AES algorithms have been discussed. In Chapter five, The AES algorithm is modified. Initially we use BCH codes and calculated secret keys for each round in AES algorithm. In second step, mixed column matrices have been computed by using BCH codes for each round. This modified AES algorithm has been used for text encryption and then applied avalanche effect to cipher text. N1ST statistical test have been applied on proposed text encryption scheme.